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Craster's last son is taken away by a white walker and the Night King takes the boy and puts his finger to his face. The baby's eyes turn the color of those of the White Walkers, bright blue.
We know they are not Wights so they are something else other than reanimated corpses. So what happens to the babies? Do they grow like normal humans? Or is the process something different and something else happen?
It is simply unknown as of now.
Others are a mysterical species who are most certainly not dead. One key aspect of life is growing up. Quoting GRRM:
The Others are not dead. They are strange, beautiful… think, oh… the Sidhe made of ice, something like that… a different sort of life… inhuman, elegant, dangerous.
Following is the compilation of available info and inferences drawn on that info from both Canons:
In Books, there is no explicit indication of what exactly do Others do to the offered children. But it is heavily implied that they make them one of them which in turn implies that somehow the Others are either incapable of reproducing or they can't reproduce fast enough to build their army for the upcoming war.
It has been 8000 years since Others were defeated. The information about them is virtually none. Even the Night's Watch has lost its knowledge about their kind. So as of now, we do not know what exactly is the process involved in making one A white Walker.
We know Craster offers his sons to the Others. From Jon III ACOK:
If it’s a girl, that’s not so bad, she’ll grown in a few years and he’d marry her. But Nella says its to be a boy, and she’s had six and knows these things. He gives the boys to the gods. Come the white cold, he does and of late it comes more often.
Since many of Craster's daughters did grow up in his life time and he did marry them, it is implied that many of his sons must have grown up as well, albeit as the Others, not as humans.
One of Craster's wives confirmed this to Sam after mutiny at Craster's keep:
The boy’s bothers… Craster’s sons. The white cold's rising out there, crow. I can feel it in my bones. These poor old bones don’t lie. They’ll be here soon, the sons.
If she says they will be here soon, that means they must have grown up. It is implied because well, toddlers can't walk or travel by themselves. If the boy's brothers are coming, they must have grown up into adults. After all, in all the encounters with Others, we haven't seen a single instance of an infant Other coming to stick his icy blade in someone's heart.
The event depicted in the Show in this regard never happened in the Books.
In show it is explicitly described that the Others turn the babies into their kind, as evident from the video linked in your own post.
As for the process involved, only thing that we see is that the Night's king just has touch a Child and convert him into a White Walker. But I don't believe only touch is required because Night's King also touched Bran but he did not turn into a White Walker.
So it appears, only infants can be converted into White Walkers, Adults have to be killed and reanimated as Wights, thralls to the Others.
It is not known if all White Walkers can convert an infant into one of them or only Night's King can do that. It is likely that only Night's King can do that because rest of the Others simply put the child down there to await his Majesty's pleasure. None of them made any attempt to convert the child and waited for their leader.
About Growth of Others, we do not know anything. But since they converted Infants, it is probable that the converted boys grow up into youth of the Others.
Another point to note is that their growth and aging process appears to be different than humans. Night's King was supposedly created during First Men-Children of the Forest wars which happened 12,000 years ago. He seems pretty fit for a 12,000+ years old guy.
It is not known if all White Walkers can convert an infant into one of them or only Night's King can do that. The other part is my own inference
Fire Consumes, Cold Preserve - Aemon Targaryen - Corolary: Absolute cold preserves absolutely.
Jul 1, 2016 at 16:11
They said in one of the episodes that the white walkers were buried under the ground for 1000 years. Then they rise again. Maybe those are the babies that will rise again in another 1000 years. It's plausible!
